HMMT 2018

For some real number $c$, the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$ ?

Compute the positive real number $x$ satisfying

$$x^{\left(2 x^{6}\right)}=3$$

There are two prime numbers $p$ so that $5 p$ can be expressed in the form $\left\lfloor\frac{n^{2}}{5}\right\rfloor$ for some positive integer $n$. What is the sum of these two prime numbers?

Distinct prime numbers $p, q, r$ satisfy the equation

$$2 p q r+50 p q=7 p q r+55 p r=8 p q r+12 q r=A$$

for some positive integer $A$. What is $A$ ?

Let $\omega_{1}, \omega_{2}, \ldots, \omega_{100}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set

$$S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}$$

Let $M$ be the maximum possible number of unique values in $S$, and let $N$ be the minimum possible number of unique values in $S$. Find $M-N$.

Let $\alpha, \beta$, and $\gamma$ be three real numbers. Suppose that

$$\begin{aligned} \cos \alpha+\cos \beta+\cos \gamma & =1 \\ \sin \alpha+\sin \beta+\sin \gamma & =1 \end{aligned}$$

Find the smallest possible value of $\cos \alpha$.

Rachel has the number 1000 in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1$. When she puts the number $x$ in her right pocket, the number changes to $x^{-1}$. Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E$, then compute $\left\lfloor\frac{E}{10}\right\rfloor$.

For how many pairs of sequences of nonnegative integers $\left(b_{1}, b_{2}, \ldots, b_{2018}\right)$ and $\left(c_{1}, c_{2}, \ldots, c_{2018}\right)$ does there exist a sequence of nonnegative integers $\left(a_{0}, \ldots, a_{2018}\right)$ with the following properties:

• For $0 \leq i \leq 2018, a_{i}<2^{2018}$;
• For $1 \leq i \leq 2018, b_{i}=a_{i-1}+a_{i}$ and $c_{i}=a_{i-1} \mid a_{i}$; where $\mid$ denotes the bitwise or operation? (The bitwise or of two nonnegative integers $x=\cdots x_{3} x_{2} x_{1} x_{0}$ and $y=\cdots y_{3} y_{2} y_{1} y_{0}$ expressed in binary is defined as $x \mid y=\cdots z_{3} z_{2} z_{1} z_{0}$, where $z_{i}=1$ if at least one of $x_{i}$ and $y_{i}$ is 1 , and 0 otherwise.)

Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\frac{1}{2} \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 2$. Find the maximum possible value of $\frac{\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right) x_{4}}{\left(x_{4}+x_{2}\right)\left(x_{4}+x_{3}\right) x_{1}}$ (over all valid choices of $\left.a, b, d\right)$.

Let $S$ be a randomly chosen 6 -element subset of the set $\{0,1,2, \ldots, n\}$. Consider the polynomial $P(x)=\sum_{i \in S} x^{i}$. Let $X_{n}$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most 3 with integer coefficients satisfying $Q(0) \neq 0$. Find the limit of $X_{n}$ as $n$ goes to infinity.

Triangle $G R T$ has $G R=5, R T=12$, and $G T=13$. The perpendicular bisector of $G T$ intersects the extension of $G R$ at $O$. Find $T O$.

Points $A, B, C, D$ are chosen in the plane such that segments $A B, B C, C D, D A$ have lengths $2,7,5$, 12 , respectively. Let $m$ be the minimum possible value of the length of segment $A C$ and let $M$ be the maximum possible value of the length of segment $A C$. What is the ordered pair $(m, M)$ ?

How many noncongruent triangles are there with one side of length 20 , one side of length 17 , and one $60^{\circ}$ angle?

A paper equilateral triangle of side length 2 on a table has vertices labeled $A, B, C$. Let $M$ be the point on the sheet of paper halfway between $A$ and $C$. Over time, point $M$ is lifted upwards, folding the triangle along segment $B M$, while $A, B$, and $C$ remain on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $A B C M$ at any time during this process.

In the quadrilateral $M A R E$ inscribed in a unit circle $\omega, A M$ is a diameter of $\omega$, and $E$ lies on the angle bisector of $\angle R A M$. Given that triangles $R A M$ and $R E M$ have the same area, find the area of quadrilateral $M A R E$.

Let $A B C$ be an equilateral triangle of side length 1. For a real number $0<x<0.5$, let $A_{1}$ and $A_{2}$ be the points on side $B C$ such that $A_{1} B=A_{2} C=x$, and let $T_{A}=\triangle A A_{1} A_{2}$. Construct triangles $T_{B}=\triangle B B_{1} B_{2}$ and $T_{C}=\triangle C C_{1} C_{2}$ similarly. There exist positive rational numbers $b, c$ such that the region of points inside all three triangles $T_{A}, T_{B}, T_{C}$ is a hexagon with area

$$\frac{8 x^{2}-b x+c}{(2-x)(x+1)} \cdot \frac{\sqrt{3}}{4}$$

Find $(b, c)$.

Triangle $A B C$ has sidelengths $A B=14, A C=13$, and $B C=15$. Point $D$ is chosen in the interior of $\overline{A B}$ and point $E$ is selected uniformly at random from $\overline{A D}$. Point $F$ is then defined to be the intersection point of the perpendicular to $\overline{A B}$ at $E$ and the union of segments $\overline{A C}$ and $\overline{B C}$. Suppose that $D$ is chosen such that the expected value of the length of $\overline{E F}$ is maximized. Find $A D$.

Let $A B C$ be an equilateral triangle with side length 8 . Let $X$ be on side $A B$ so that $A X=5$ and $Y$ be on side $A C$ so that $A Y=3$. Let $Z$ be on side $B C$ so that $A Z, B Y, C X$ are concurrent. Let $Z X, Z Y$ intersect the circumcircle of $A X Y$ again at $P, Q$ respectively. Let $X Q$ and $Y P$ intersect at $K$. Compute $K X \cdot K Q$.

Po picks 100 points $P_{1}, P_{2}, \ldots, P_{100}$ on a circle independently and uniformly at random. He then draws the line segments connecting $P_{1} P_{2}, P_{2} P_{3}, \ldots, P_{100} P_{1}$. When all of the line segments are drawn, the circle is divided into a number of regions. Find the expected number of regions that have all sides bounded by straight lines.

Let $A B C$ be a triangle such that $A B=6, B C=5, A C=7$. Let the tangents to the circumcircle of $A B C$ at $B$ and $C$ meet at $X$. Let $Z$ be a point on the circumcircle of $A B C$. Let $Y$ be the foot of the perpendicular from $X$ to $C Z$. Let $K$ be the intersection of the circumcircle of $B C Y$ with line $A B$. Given that $Y$ is on the interior of segment $C Z$ and $Y Z=3 C Y$, compute $A K$.

Consider a $2 \times 3$ grid where each entry is one of 0,1 , and 2 . For how many such grids is the sum of the numbers in every row and in every column a multiple of 3 ? One valid grid is shown below.

$$\left[\begin{array}{lll} 1 & 2 & 0 \\ 2 & 1 & 0 \end{array}\right]$$

Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a<b$ and there are no other five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b-a$ ? (A number is a palindrome if it reads the same forwards and backwards in base 10.)

A $4 \times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side.

How many ways are there for Nick to travel from $(0,0)$ to $(16,16)$ in the coordinate plane by moving one unit in the positive $x$ or $y$ direction at a time, such that Nick changes direction an odd number of times?

A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of the number of total marbles he drew?

Sarah stands at $(0,0)$ and Rachel stands at $(6,8)$ in the Euclidean plane. Sarah can only move 1 unit in the positive $x$ or $y$ direction, and Rachel can only move 1 unit in the negative $x$ or $y$ direction. Each second, Sarah and Rachel see each other, independently pick a direction to move at the same time, and move to their new position. Sarah catches Rachel if Sarah and Rachel are ever at the same point. Rachel wins if she is able to get to $(0,0)$ without being caught; otherwise, Sarah wins. Given that both of them play optimally to maximize their probability of winning, what is the probability that Rachel wins?

A tourist is learning an incorrect way to sort a permutation $\left(p_{1}, \ldots, p_{n}\right)$ of the integers $(1, \ldots, n)$. We define a $f i x$ on two adjacent elements $p_{i}$ and $p_{i+1}$, to be an operation which swaps the two elements if $p_{i}>p_{i+1}$, and does nothing otherwise. The tourist performs $n-1$ rounds of fixes, numbered $a=1,2, \ldots, n-1$. In round $a$ of fixes, the tourist fixes $p_{a}$ and $p_{a+1}$, then $p_{a+1}$ and $p_{a+2}$, and so on, up to $p_{n-1}$ and $p_{n}$. In this process, there are $(n-1)+(n-2)+\cdots+1=\frac{n(n-1)}{2}$ total fixes performed. How many permutations of $(1, \ldots, 2018)$ can the tourist start with to obtain $(1, \ldots, 2018)$ after performing these steps?

A permutation of $\{1,2, \ldots, 7\}$ is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ and $[6,5,7]$, since when these blocks are sorted, the permutation becomes $(1,2,3,4,5,6,7)$. Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly.

How many ordered sequences of 36 digits have the property that summing the digits to get a number and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits range from 0 to 9. )

Lily has a $300 \times 300$ grid of squares. She now removes $100 \times 100$ squares from each of the four corners and colors each of the remaining 50000 squares black and white. Given that no $2 \times 2$ square is colored in a checkerboard pattern, find the maximum possible number of (unordered) pairs of squares such that one is black, one is white and the squares share an edge.

In an $n \times n$ square array of $1 \times 1$ cells, at least one cell is colored pink. Show that you can always divide the square into rectangles along cell borders such that each rectangle contains exactly one pink cell.

Is the number

$$\left(1+\frac{1}{2}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{6}\right) \ldots\left(1+\frac{1}{2018}\right)$$

greater than, less than, or equal to 50 ?

Michelle has a word with $2^{n}$ letters, where a word can consist of letters from any alphabet. Michelle performs a switcheroo on the word as follows: for each $k=0,1, \ldots, n-1$, she switches the first $2^{k}$ letters of the word with the next $2^{k}$ letters of the word. For example, for $n=3$, Michelle changes

$$A B C D E F G H \rightarrow B A C D E F G H \rightarrow C D B A E F G H \rightarrow E F G H C D B A$$

in one switcheroo. In terms of $n$, what is the minimum positive integer $m$ such that after Michelle performs the switcheroo operation $m$ times on any word of length $2^{n}$, she will receive her original word?

In acute triangle $A B C$, let $D, E$, and $F$ be the feet of the altitudes from $A, B$, and $C$ respectively, and let $L, M$, and $N$ be the midpoints of $B C, C A$, and $A B$, respectively. Lines $D E$ and $N L$ intersect at $X$, lines $D F$ and $L M$ intersect at $Y$, and lines $X Y$ and $B C$ intersect at $Z$. Find $\frac{Z B}{Z C}$ in terms of $A B, A C$, and $B C$.

Is it possible for the projection of the set of points $(x, y, z)$ with $0 \leq x, y, z \leq 1$ onto some two-dimensional plane to be a simple convex pentagon?

Let $n \geq 2$ be a positive integer. A subset of positive integers $S$ is said to be comprehensive if for every integer $0 \leq x<n$, there is a subset of $S$ whose sum has remainder $x$ when divided by $n$. Note that the empty set has sum 0 . Show that if a set $S$ is comprehensive, then there is some (not necessarily proper) subset of $S$ with at most $n-1$ elements which is also comprehensive.

Let $[n]$ denote the set of integers $\{1,2, \ldots, n\}$. We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \geq 1$ such that $f^{b}(1)=a$ and $f^{c}(a)=1$. $\left(f^{k}(x)\right.$ denotes the result of applying $f$ to $x k$ times).

Allen plays a game on a tree with $2 n$ vertices, each of whose vertices can be red or blue. Initially, all of the vertices of the tree are colored red. In one move, Allen is allowed to take two vertices of the same color which are connected by an edge and change both of them to the opposite color. He wins if at any time, all of the vertices of the tree are colored blue. (a) (20) Show that Allen can win if and only if the vertices can be split up into two groups $V_{1}$ and $V_{2}$ of size $n$, such that each edge in the tree has one endpoint in $V_{1}$ and one endpoint in $V_{2}$. (b) (40) Let $V_{1}=\left\{a_{1}, \ldots, a_{n}\right\}$ and $V_{2}=\left\{b_{1}, \ldots, b_{n}\right\}$ from part (a). Let $M$ be the minimum over all permutations $\sigma$ of $\{1, \ldots, n\}$ of the quantity

$$\sum_{i=1}^{n} d\left(a_{i}, b_{\sigma(i)}\right)$$

where $d(v, w)$ denotes the number of edges along the shortest path between vertices $v$ and $w$ in the tree. Show that if Allen can win, then the minimum number of moves that it can take for Allen to win is equal to $M$. (A graph consists of a set of vertices and some edges between distinct pairs of vertices. It is connected if every pair of vertices are connected by some path of one or more edges. A tree is a graph which is connected, in which the number of edges is one less than the number of vertices.)

Evan has a simple graph with $v$ vertices and $e$ edges. Show that he can delete at least $\frac{e-v+1}{2}$ edges so that each vertex still has at least half of its original degree.

Let $n$ and $m$ be positive integers which are at most $10^{10}$. Let $R$ be the rectangle with corners at $(0,0),(n, 0),(n, m),(0, m)$ in the coordinate plane. A simple non-self-intersecting quadrilateral with vertices at integer coordinates is called far-reaching if each of its vertices lie on or inside $R$, but each side of $R$ contains at least one vertex of the quadrilateral. Show that there is a far-reaching quadrilateral with area at most $10^{6}$. (A side of a rectangle includes the two endpoints.)

A square can be divided into four congruent figures as shown:

If each of the congruent figures has area 1 , what is the area of the square?

John has a 1 liter bottle of pure orange juice. He pours half of the contents of the bottle into a vat, fills the bottle with water, and mixes thoroughly. He then repeats this process 9 more times. Afterwards, he pours the remaining contents of the bottle into the vat. What fraction of the liquid in the vat is now water?

Allen and Yang want to share the numbers $1,2,3,4,5,6,7,8,9,10$. How many ways are there to split all ten numbers among Allen and Yang so that each person gets at least one number, and either Allen's numbers or Yang's numbers sum to an even number?

Find the sum of the digits of $11 \cdot 101 \cdot 111 \cdot 110011$.

Randall proposes a new temperature system called Felsius temperature with the following conversion between Felsius ${ }^{\circ} E$, Celsius ${ }^{\circ} C$, and Fahrenheit ${ }^{\circ} F$ :

$${ }^{\circ} E=\frac{7 \times{ }^{\circ} C}{5}+16=\frac{7 \times{ }^{\circ} F-80}{9} .$$

For example, $0^{\circ} C=16^{\circ} E$. Let $x, y, z$ be real numbers such that $x^{\circ} C=x^{\circ} E, y^{\circ} E=y^{\circ} F, z^{\circ} C=$ $z^{\circ} F$. Find $x+y+z$.

A bug is on a corner of a cube. A healthy path for the bug is a path along the edges of the cube that starts and ends where the bug is located, uses no edge multiple times, and uses at most two of the edges adjacent to any particular face. Find the number of healthy paths.

A triple of integers $(a, b, c)$ satisfies $a+b c=2017$ and $b+c a=8$. Find all possible values of $c$.

Suppose a real number $x>1$ satisfies

$$\log _{2}\left(\log _{4} x\right)+\log _{4}\left(\log _{16} x\right)+\log _{16}\left(\log _{2} x\right)=0$$

Compute

$$\log _{2}\left(\log _{16} x\right)+\log _{16}\left(\log _{4} x\right)+\log _{4}\left(\log _{2} x\right)$$

In a game, $N$ people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let $m$ be the maximum possible number such that it is possible to win the game by writing down $m$. Find the smallest possible value of $N$ for which it is possible to win the game by writing down $m$ in a room of $N$ people.

Let a positive integer $n$ be called a cubic square if there exist positive integers $a, b$ with $n=$ $\operatorname{gcd}\left(a^{2}, b^{3}\right)$. Count the number of cubic squares between 1 and 100 inclusive.

FInd the value of

$$\sum_{k=1}^{60} \sum_{n=1}^{k} \frac{n^{2}}{61-2 n}$$

$\triangle P N R$ has side lengths $P N=20, N R=18$, and $P R=19$. Consider a point $A$ on $P N . \triangle N R A$ is rotated about $R$ to $\triangle N^{\prime} R A^{\prime}$ so that $R, N^{\prime}$, and $P$ lie on the same line and $A A^{\prime}$ is perpendicular to $P R$. Find $\frac{P A}{A N}$.

Suppose $\triangle A B C$ has lengths $A B=5, B C=8$, and $C A=7$, and let $\omega$ be the circumcircle of $\triangle A B C$. Let $X$ be the second intersection of the external angle bisector of $\angle B$ with $\omega$, and let $Y$ be the foot of the perpendicular from $X$ to $B C$. Find the length of $Y C$.

Given that $x$ is a positive real, find the maximum possible value of

$$\sin \left(\tan ^{-1}\left(\frac{x}{9}\right)-\tan ^{-1}\left(\frac{x}{16}\right)\right)$$

Michael picks a random subset of the complex numbers $\left\{1, \omega, \omega^{2}, \ldots, \omega^{2017}\right\}$ where $\omega$ is a primitive $2018^{\text {th }}$ root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is $S$, what is the expected value of $|S|^{2}$ ? (The sum of the elements of the empty set is 0 .)

Solve for $x$ :

$$x\lfloor x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor\rfloor=122 .$$

Compute the value of

$$\frac{\cos 30.5^{\circ}+\cos 31.5^{\circ}+\ldots+\cos 44.5^{\circ}}{\sin 30.5^{\circ}+\sin 31.5^{\circ}+\ldots+\sin 44.5^{\circ}}$$

Compute the number of integers $n \in\{1,2, \ldots, 300\}$ such that $n$ is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths.

Suppose there are 100 cookies arranged in a circle, and 53 of them are chocolate chip, with the remainder being oatmeal. Pearl wants to choose a contiguous subsegment of exactly 67 cookies and wants this subsegment to have exactly $k$ chocolate chip cookies. Find the sum of the $k$ for which Pearl is guaranteed to succeed regardless of how the cookies are arranged.

Triangle $\triangle A B C$ has $A B=21, B C=55$, and $C A=56$. There are two points $P$ in the plane of $\triangle A B C$ for which $\angle B A P=\angle C A P$ and $\angle B P C=90^{\circ}$. Find the distance between them.

You are the first lucky player to play in a slightly modified episode of Deal or No Deal! Initially, there are sixteen cases marked 1 through 16. The dollar amounts in the cases are the powers of 2 from $2^{1}=2$ to $2^{16}=65536$, in some random order. The game has eight turns. In each turn, you choose a case and claim it, without opening it. Afterwards, a random remaining case is opened and revealed to you, then removed from the game. At the end of the game, all eight of your cases are revealed and you win all of the money inside them. However, the hosts do not realize you have X-ray vision and can see the amount of money inside each case! What is the expected amount of money you will make, given that you play optimally?

How many graphs are there on 10 vertices labeled $1,2, \ldots, 10$ such that there are exactly 23 edges and no triangles?

Kevin starts with the vectors $(1,0)$ and $(0,1)$ and at each time step, he replaces one of the vectors with their sum. Find the cotangent of the minimum possible angle between the vectors after 8 time steps.

Find the largest positive integer $n$ for which there exist $n$ finite sets $X_{1}, X_{2}, \ldots, X_{n}$ with the property that for every $1 \leq a<b<c \leq n$, the equation

$$\left|X_{a} \cup X_{b} \cup X_{c}\right|=\lceil\sqrt{a b c}\rceil$$

holds.

Fran writes the numbers $1,2,3, \ldots, 20$ on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number $n$ uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors of $n$ that are still on the chalkboard (including $n$ itself). What is the expected number of moves that Fran must make to erase all the numbers?

Let $A B C$ be a triangle with $\angle A=18^{\circ}, \angle B=36^{\circ}$. Let $M$ be the midpoint of $A B, D$ a point on ray $C M$ such that $A B=A D ; E$ a point on ray $B C$ such that $A B=B E$, and $F$ a point on ray $A C$ such that $A B=A F$. Find $\angle F D E$.

There are 2018 frogs in a pool and there is 1 frog on the shore. In each time-step thereafter, one random frog moves position. If it was in the pool, it jumps to the shore, and vice versa. Find the expected number of time-steps before all frogs are in the pool for the first time.

Arnold and Kevin are playing a game in which Kevin picks an integer $1 \leq m \leq 1001$, and Arnold is trying to guess it. On each turn, Arnold first pays Kevin 1 dollar in order to guess a number $k$ of Arnold's choice. If $m \geq k$, the game ends and he pays Kevin an additional $m-k$ dollars (possibly zero). Otherwise, Arnold pays Kevin an additional 10 dollars and continues guessing. Which number should Arnold guess first to ensure that his worst-case payment is minimized?

Let $a, b, c$ be positive integers. All the roots of each of the quadratics

$$a x^{2}+b x+c, a x^{2}+b x-c, a x^{2}-b x+c, a x^{2}-b x-c$$

are integers. Over all triples $(a, b, c)$, find the triple with the third smallest value of $a+b+c$.

Find the number of unordered pairs $\{a, b\}$, where $a, b \in\{0,1,2, \ldots, 108\}$ such that 109 divides $a^{3}+b^{3}-a b$.

In triangle $A B C, A B=6, B C=7$ and $C A=8$. Let $D, E, F$ be the midpoints of sides $B C$, $A C, A B$, respectively. Also let $O_{A}, O_{B}, O_{C}$ be the circumcenters of triangles $A F D, B D E$, and $C E F$, respectively. Find the area of triangle $O_{A} O_{B} O_{C}$.

How many 48-tuples of positive integers $\left(a_{1}, a_{2}, \ldots, a_{48}\right)$ between 0 and 100 inclusive have the property that for all $1 \leq i<j \leq 48, a_{i} \notin\left\{a_{j}, a_{j}+1\right\}$ ?

679 contestants participated in HMMT February 2017. Let $N$ be the number of these contestants who performed at or above the median score in at least one of the three individual tests. Estimate $N$. An estimate of $E$ earns $\left\lfloor 20-\frac{|E-N|}{2}\right\rfloor$ or 0 points, whichever is greater.

The integers $1,2,3,4,5,6,7,8,9,10$ are written on a blackboard. Each day, a teacher chooses one of the integers uniformly at random and decreases it by 1 . Let $X$ be the expected value of the number of days which elapse before there are no longer positive integers on the board. Estimate $X$. An estimate of $E$ earns $\left\lfloor 20 \cdot 2^{-|X-E| / 8}\right\rfloor$ points.

In a wooden block shaped like a cube, all the vertices and edge midpoints are marked. The cube is cut along all possible planes that pass through at least four marked points. Let $N$ be the number of pieces the cube is cut into. Estimate $N$. An estimate of $E>0$ earns $\lfloor 20 \min (N / E, E / N)\rfloor$ points.

In the game of Connect Four, there are seven vertical columns which have spaces for six tokens. These form a $7 \times 6$ grid of spaces. Two players White and Black move alternately. A player takes a turn by picking a column which is not already full and dropping a token of their color into the lowest unoccupied space in that column. The game ends when there are four consecutive tokens of the same color in a line, either horizontally, vertically, or diagonally. The player who has four tokens in a row of their color wins. Assume two players play this game randomly. Each player, on their turn, picks a random column which is not full and drops a token of their color into that column. This happens until one player wins or all of the columns are filled. Let $P$ be the probability that all of the columns are filled without any player obtaining four tokens in a row of their color. Estimate $P$. An estimate of $E>0$ earns $\lfloor 20 \min (P / E, E / P)\rfloor$ points.